{
  "abstract": "We define a generalized monoid of words, an asymptotic quotient of moderate nets of finite words, to regularize divergent products of arithmetic origin. Divergent sequences are encoded as moderate nets over a finite alphabet, while prefix-dependent observables are evaluated along admissible asymptotic scales. We characterize the evaluation and renormalization schemes axiomatically. Once the asymptotic scale and class of admissible regular expansions are fixed, the constant-term prescription is uniquely determined. Thus the regularized value is canonical relative to the chosen scale and evaluation functional, without absolute uniqueness. The regularizing functionals arise as expectations of logarithmic-prefix functionals under empirical distributions of sign oscillations. The alternating product of integers is regularized to $\\sqrt{2/π}$, matching zeta regularization, while the alternating product of factorials yields $(2/π)^{1/4}$ and satisfies $\\mathfrak{F}_{\\mathrm{ren}}=\\sqrt{P_{\\mathrm{ren}}}$. Products governed by dyadic block signs acquire finite values through logarithmic-scale averaging in the corresponding dyadic quotient, invariant under dyadic equivalence. For the Thue--Morse sequence, zeta regularization is well defined via meromorphic continuation of its Dirichlet series. A separate convergent functional associated with its logarithmic partial sums yields $\\exp(\\mathcal F)\\approx0.9732$. We also study numerically a substitution-generated sign sequence with exact square-root growth on dyadic blocks and fractal log-periodic structure. Its Dirichlet series shows a candidate log-periodic pole structure, separating local scale constants from a candidate global regularized value. The choice of asymptotic scale is determined by the arithmetic structure of the sequence: logarithmic scales for ordinary alternating products and dyadic scales for block-automatic sequences.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.06297",
  "author_total": 2,
  "authors": [
    {
      "name": "A. Alvarez Cruz"
    },
    {
      "name": "E. A. Alvarez Gutierrez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5qwpzxpnqtttt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.06297",
  "fulltext_url": null,
  "id": "5qwpzxpnqtttt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.06297",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7201877560",
  "pdf_url": "https://arxiv.org/pdf/2608.06297",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5qwpzxpnqtttt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2608.06297",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.06297"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Asymptotic monoids of words and logarithmic-prefix functionals for divergent arithmetic products",
  "title_text": "Asymptotic monoids of words and logarithmic-prefix functionals for divergent arithmetic products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/5qwpzxpnqtttt",
  "venue": "arXiv",
  "versions": []
}